Tom Daschle was to become the new secretary of health, and Nancy Killefer was Mr Obama's choice to oversee budget and spending reform.
They both withdrew when questions were raised about their tax affairs.
The president said he was taking full responsibility and promised it would not happen again.
But a BBC Washington correspondent says the growing number of Mr Obama's nominees facing tax problems appears to be undermining his calls for a new era of responsibility.
Speaking in a series of TV interviews late on Tuesday, President Obama also said he wanted to avoid including anything that might spark a trade war in a planned economic stimulus package.
Mr Obama was responding to EU and Canadian criticism of a "Buy American" clause in the $800bn (£567bn) package, which is being debated in the Senate.
Mr Obama said he did not want the US "to start sending a message that somehow we're just looking after ourselves and not concerned with world trade".
Distraction fears
Mr Daschle, a former senator and an early Obama backer, withdrew his nomination on Tuesday, after it was revealed last week that he had failed to pay some $130,000 in taxes.
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President Obama: 'We don't have two sets of rules'
He said he would have not been able to operate "with the full faith of Congress and the American people".
He also said he did not want to "be a distraction" from Mr Obama's agenda.
The president said he had accepted his nominee's withdrawal "with sadness and regret".
Mr Daschle's move came only hours after Nancy Killefer, nominated as budget watchdog, also withdrew over tax issues.